math.sqrt on complex, imaginary part, im greater than 0 branch

Percentage Accurate: 41.4% → 88.1%

Time: 8.8s

Alternatives: 5

Speedup: 2.0×

Specification

\[im > 0\]

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 41.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.4e-27)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.4e-27) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.4e-27) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.4e-27:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.4e-27)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.4e-27)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.4e-27], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.4e-27

   1. Initial program 51.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. sub-neg 51.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + \left(-re\right)\right)}} \]

      2. sqr-neg 51.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{\left(-im\right) \cdot \left(-im\right)}} + \left(-re\right)\right)} \]

      3. sub-neg 51.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + \left(-im\right) \cdot \left(-im\right)} - re\right)}} \]

      4. sqr-neg 51.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}} - re\right)} \]

      5. hypot-define 94.8%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 94.8%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
   4. Add Preprocessing

   if 1.4e-27 < re

   1. Initial program 11.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf 54.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 54.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{{im}^{2}}{re}}\right)} \]

      2. *-commutative 54.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{{im}^{2}}{re}} \cdot 1\right)} \]

      3. sqrt-div 68.9%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\frac{\sqrt{{im}^{2}}}{\sqrt{re}}} \cdot 1\right) \]

      4. sqrt-pow1 82.5%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{{im}^{\left(\frac{2}{2}\right)}}}{\sqrt{re}} \cdot 1\right) \]

      5. metadata-eval 82.5%

         \[\leadsto 0.5 \cdot \left(\frac{{im}^{\color{blue}{1}}}{\sqrt{re}} \cdot 1\right) \]

      6. pow1 82.5%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{im}}{\sqrt{re}} \cdot 1\right) \]

   5. Applied egg-rr 82.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{im}{\sqrt{re}} \cdot 1\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity 82.5%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   7. Simplified 82.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e-55)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2e-29)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-55) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2e-29) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.4d-55)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2d-29) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-55) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2e-29) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.4e-55:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2e-29:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.4e-55)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2e-29)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.4e-55)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2e-29)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.4e-55], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2e-29], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.39999999999999991e-55

   1. Initial program 45.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around -inf 75.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 75.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 75.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -2.39999999999999991e-55 < re < 1.99999999999999989e-29

   1. Initial program 56.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 83.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)} \]

   if 1.99999999999999989e-29 < re

   1. Initial program 11.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf 54.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 54.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{{im}^{2}}{re}}\right)} \]

      2. *-commutative 54.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{{im}^{2}}{re}} \cdot 1\right)} \]

      3. sqrt-div 68.9%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\frac{\sqrt{{im}^{2}}}{\sqrt{re}}} \cdot 1\right) \]

      4. sqrt-pow1 82.5%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{{im}^{\left(\frac{2}{2}\right)}}}{\sqrt{re}} \cdot 1\right) \]

      5. metadata-eval 82.5%

         \[\leadsto 0.5 \cdot \left(\frac{{im}^{\color{blue}{1}}}{\sqrt{re}} \cdot 1\right) \]

      6. pow1 82.5%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{im}}{\sqrt{re}} \cdot 1\right) \]

   5. Applied egg-rr 82.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{im}{\sqrt{re}} \cdot 1\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity 82.5%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   7. Simplified 82.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.2e-57)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.15e-27) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-57) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.15e-27) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-6.2d-57)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.15d-27) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-57) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.15e-27) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.2e-57:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.15e-27:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.2e-57)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.15e-27)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.2e-57)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.15e-27)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.2e-57], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.15e-27], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.19999999999999952e-57

   1. Initial program 45.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around -inf 75.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 75.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 75.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -6.19999999999999952e-57 < re < 1.15e-27

   1. Initial program 56.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 83.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   4. Step-by-step derivation

      1. *-commutative 83.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   5. Simplified 83.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   if 1.15e-27 < re

   1. Initial program 11.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf 54.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 54.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{{im}^{2}}{re}}\right)} \]

      2. *-commutative 54.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{{im}^{2}}{re}} \cdot 1\right)} \]

      3. sqrt-div 68.9%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\frac{\sqrt{{im}^{2}}}{\sqrt{re}}} \cdot 1\right) \]

      4. sqrt-pow1 82.5%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{{im}^{\left(\frac{2}{2}\right)}}}{\sqrt{re}} \cdot 1\right) \]

      5. metadata-eval 82.5%

         \[\leadsto 0.5 \cdot \left(\frac{{im}^{\color{blue}{1}}}{\sqrt{re}} \cdot 1\right) \]

      6. pow1 82.5%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{im}}{\sqrt{re}} \cdot 1\right) \]

   5. Applied egg-rr 82.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{im}{\sqrt{re}} \cdot 1\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity 82.5%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   7. Simplified 82.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.2 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.2e-56) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.2e-56) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.2d-56)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.2e-56) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.2e-56:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.2e-56)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.2e-56)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.2e-56], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.2e-56

   1. Initial program 45.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around -inf 75.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 75.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 75.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.2e-56 < re

   1. Initial program 37.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 56.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   4. Step-by-step derivation

      1. *-commutative 56.6%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   5. Simplified 56.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.2 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot im} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (sqrt (* 2.0 im))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * im))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * im));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * im))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * im));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.1%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
2. Add Preprocessing

3. Taylor expanded in re around 0 48.2%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
4. Step-by-step derivation

   1. *-commutative 48.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

5. Simplified 48.2%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

6. Final simplification 48.2%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))